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Varvara68 [4.7K]
3 years ago
9

Randy presses RAND on his calculator twice to obtain two random numbers between 0 and 1. Let $p$ be the probability that these t

wo numbers and 1 form the sides of an obtuse triangle. Find $4p$.
Mathematics
2 answers:
Ludmilka [50]3 years ago
7 0

Answer:

pi - 2 would be the answer to AoPs.

anygoal [31]3 years ago
6 0

Answer:

\frac{\pi}{4}

Step-by-step explanation:

Lets call x,y the numbers we obtain from the calculator. x and y are independent random variables of uniform[0,1] distribution.

Lets note that, since both x and y are between 0 and 1, then 1 is the biggest side of the triangle.

Lets first make a geometric interpretation. If the triangle were to be rectangle, then the side of lenght 1 should be its hypotenuse, and therefore x and y should satisfy this property:

x²+y² = 1

Remember that in this case we are supposing the triangle to be rectangle. But the exercise asks us to obtain an obtuse triangle. For that we will need to increase the angle obtained by the sides of lenght x and y. We can do that by 'expanding' the triangle, but if we do that preserving the values of x and y, then the side of lenght 1 should increase its lenght, which we dont want to. Thus, if we expand the triangle then we should also reduce the value of x and/or y so that the side of lenght 1 could preserve its lenght. With this intuition we could deduce that

x²+y² < 1

Now lets do a more mathematical approach. According to the Cosine theorem, a triangle of three sides of lenght a,b,c satisfies

a² = b²+c² - 2bc* cos(α), where α is the angle between the sides of lenght b and c.

Aplying this formula to our triangle, we have that

1^2 = x^2 + y^2 - 2bc* cos(\alpha) , where \alpha is the angle between the sides of lenght x and y.

Since the triangle is obtuse, then \pi/2 <  \alpha < \pi , and for those values cos(\alpha) is negative , hence we also obtain

1 > x² + y²

Thus, we need to calculate P(x²+y² <1). This probability can be calculated throught integration. We need to use polar coordinates.

(x, y) = (r*cosФ,r*senФ)

Where r is between 0 and 1, and Ф is between 0 and \pi /2 (that way, the numbers are positive).

The jacobian matrix has determinant r, therefore,

{\int\int}\limits_{x^2+y^2 < 1}  \, dxdy = \int\limits^1_0\int\limits^{\frac{\pi}{2}}_0 {r} \, d\phi dr = \frac{\pi}{2} * \int\limits^1_0 {r} \, dr =    \frac{\pi}{2} * (\frac{r^2}{2} |^1_0) = \frac{\pi}{4}

As a conclusion, the probability of obtaining an obtuse triangle is \frac{\pi}{4} .

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A square has side length of 9 in. If the area is doubled, what happens to the side length?
Alika [10]

Answer:

The side length is multiplied by \sqrt{2}

Step-by-step explanation:

we know that

The area of the original square is equal to

A=9^{2}=81\ in^{2}

If the area is doubled

then

The area of the larger square is

A1=(2)81=162\ in^{2}

Remember that

If two figures are similar, then the ratio of its areas is equal to the scale factor squared

Let

z ----> the scale factor

x ----> the area of the larger square

y ---> the area of the original square

so

z^{2}=\frac{x}{y}

we have

x=162\ in\^{2}

y=81\ in\^{2}

z^{2}=\frac{162}{81}

z^{2}=2

z=\sqrt{2} ------> scale factor

therefore

The side length is multiplied by \sqrt{2}

7 0
2 years ago
Which exponential function grows at a faster rate than the quadratic function for zero is less than X is less than three
Ann [662]

Answer:

43.35 years

why?

From the above question, we are to find Time t for compound interest

The formula is given as :

t = ln(A/P) / n[ln(1 + r/n)]

A = $2500

P = Principal = $200

R = 6%

n = Compounding frequency = 1

First, convert R as a percent to r as a decimal

r = R/100

r = 6/100

r = 0.06 per year,

Then, solve the equation for t

t = ln(A/P) / n[ln(1 + r/n)]

t = ln(2,500.00/200.00) / ( 1 × [ln(1 + 0.06/1)] )

t = ln(2,500.00/200.00) / ( 1 × [ln(1 + 0.06)] )

t = 43.346 years

(credit to VmariaS)

5 0
2 years ago
Jen has 1 chocolate bar she is sharing with her friend, Joe. Jen takes 2/3 of Joe’s piece. What’s Joe’s current piece?
lana66690 [7]

Answer:1/3

Step-by-step explanation:

If Jen has 1 chocolate bar means that he has 3/3 whole .If he gives 2/3 then he will be left with 1/3 part of the chocolate bar.

3 0
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ZanzabumX [31]

Answer:

56

Step-by-step explanation:

8 0
2 years ago
Read 2 more answers
If f(x)=3x² - 4 and g(x)=x+2, find (f - g)(x).​
nika2105 [10]

Answer:

D

Step-by-step explanation:

(f - g)(x)

= f(x) - g(x)

= 3x² - 4 - (x + 2) ← distribute parenthesis by - 1

= 3x² - 4 - x - 2 ← collect like terms

= 3x² - x - 6

3 0
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